Philip Holmes

THE DYNAMICS OF COHERENT STRUCTURES IN THE WALL REGION OF A TURBULENT BOUNDARY LAYER

We have modelled the wall region of a turbulent boundary layer by expanding the instantaneous field in so-called empirical eigenfunctions, as permitted by the proper orthogonal decomposition theorem (Lumley 1967, 1981). We truncate the representation to obtain low-dimensional sets of ordinary differential equations, from the Navier-Stokes equations, via Galerkin projection. The experimentally determined eigenfunctions of Herzog (1986) are used ; these are in the form of streamwise rolls. Our model equations represent the dynamical behaviour of these rolls. We show that these equations exhibit intermittency, which we analyse using the methods of dynamical systems theory, as well as a chaotic regime. We argue that this behaviour captures major aspects of the ejection and bursting events associated with streamwise vortex pairs which have been observed in experimental work (Kline et al. 1967). We show that although this bursting behaviour is produced autonomously in the wall region, and the structure and duration of the bursts is determined there, the pressure signal from the outer part of the boundary layer triggers the bursts, and determines their average frequency. The analysis and conclusions drawn in this paper appear to be among the first to provide a reasonably coherent link between low-dimensional chaotic dynamics and a realistic turbulent open flow system.

The Physics of Optimal Decision Making: A Formal Analysis of Models of Performance in Two-Alternative Forced-Choice Tasks.

In this article, the authors consider optimal decision making in two-alternative forced-choice (TAFC) tasks. They begin by analyzing 6 models of TAFC decision making and show that all but one can be reduced to the drift diffusion model, implementing the statistically optimal algorithm (most accurate for a given speed or fastest for a given accuracy). They prove further that there is always an optimal trade-off between speed and accuracy that maximizes various reward functions, including reward rate (percentage of correct responses per unit time), as well as several other objective functions, including ones weighted for accuracy. They use these findings to address empirical data and make novel predictions about performance under optimality.

A periodically forced piecewise linear oscillator

A single-degree of freedom non-linear oscillator is considered. The non-linearity is in the restoring force and is piecewise linear with a single change in slope. Such oscillators provide models for mechanical systems in which components make intermittent contact. A limiting case in which one slope approaches infinity, an impact oscillator, is also considered. Harmonic, subharmonic, and chaotic motions are found to exist and the bifurcations leading to them are analyzed.

A nonlinear oscillator with a strange attractor

We study the nonlinear oscillator ẍ + δẋ — βx + αx3 = f cos (ωt) (A) from a qualitative viewpoint, concentrating on the bifurcational behaviour occurring as f > 0 increases for α, β, δ, ω to fixed > 0. In particular, we study the global nature of attracting motions arising as a result of bifurcations. We find that, for small and for large f, the behaviour is much as expected and that the conventional Krylov-Bogoliubov averaging theorem yields acceptable results. However, for a wide range of moderate f extremely complicated non-periodic motions arise. Such motions are called strange attractors or chaotic oscillations and have been detected in previous studies of autonomous o.d.es of dimension > 3. In the present case they are intimately connected with homoclinic orbits arising as a result of global bifurcations. We use recent results of Mel’nikov and others to prove that such motions occur in (A) and we study their structure by means of the Poincare map associated with (A). Using analogue and digital computer simulations, we provide a fairly complete characterization of the strange attractor arising for moderate f. This ergodic motion arises naturally from the deterministic differential equation (A).

The nature of the coupling between segmental oscillators of the lamprey spinal generator for locomotion: A mathematical model

We present a theoretical model which is used to explain the intersegmental coordination of the neural networks responsible for generating locomotion in the isolated spinal cord of lamprey.A simplified mathematical model of a limit cycle oscillator is presented which consists of only a single dependent variable, the phase θ(t). By coupling N such oscillators together we are able to generate stable phase locked motions which correspond to traveling waves in the spinal cord, thus simulating “fictive swimming”. We are also able to generate irregular “drifting” motions which are compared to the experimental data obtained from cords with selective surgical lesions.

A magnetoelastic strange attractor

Abstract Experimental evidence is presented for chaotic type non-periodic motions of a deterministic magnetoelastic oscillator. These motions are analogous to solutions in non-linear dynamic systems possessing what have been called “strange attractors”. In the experiments described below a ferromagnetic beam buckled between two magnets undergoes forced oscillations. Although the applied force is sinusoidal, nevertheless bounded, non-periodic, apparently chaotic motions result due to jumps between two or three stable equilibrium positions. A frequency analysis of the motion shows a broad spectrum of frequencies below the driving frequency. Also the distribution of zero crossing times shows a broad spectrum of times greater than the forcing period. The driving amplitude and frequency parameters required for these non-periodic motions are determined experimentally. A continuum model based on linear elastic and non-linear magnetic forces is developed and it is shown that this can be reduced to a single degree of freedom oscillator which exhibits chaotic solutions very similar to those observed experimentally. Thus, both experimental and theoretical evidence for the existence of a strange attractor in a deterministic dynamical system is presented.

On the Phase Reduction and Response Dynamics of Neural Oscillator Populations

We undertake a probabilistic analysis of the response of repetitively firing neural populations to simple pulselike stimuli. Recalling and extending results from the literature, we compute phase response curves (PRCs) valid near bifurcations to periodic firing for Hindmarsh-Rose, Hodgkin-Huxley, Fitz Hugh-Nagumo, and Morris-Lecar models, encompassing the four generic (codimension one) bifurcations. Phase density equations are then used to analyze the role of the bifurcation, and the resulting PRC, in responses to stimuli. In particular, we explore the interplay among stimulus duration, baseline firing frequency, and population-level response patterns. We interpret the results in terms of the signal processing measure of gain and discuss further applications and experimentally testable predictions.

The dynamics of repeated impacts with a sinusoidally vibrating table

A deceptively simple difference equation is derived which approximately describes the motion of a small ball bouncing vertically on a massive sinusoidally vibrating plate. In the case of perfect elastic impacts, the equation reduces to the “standard mapping” which has been extensively studied by physicists in connection with the motions of particles constrained in potential wells. It is shown that, for sufficiently large excitation velocities and a coefficient of restitution close to one, this deterministic dynamical system exhibits large families of irregular non-periodic solutions in addition to the expected harmonic and subharmonic motions. The physical significance of these and other chaotic motions which appear to occur frequently in non-linear oscillations is discussed.

A partial differential equation with infinitely many periodic orbits: Chaotic oscillations of a forced beam

This paper delineates a class of time-periodically perturbed evolution equations in a Banach space whose associated Poincaré map contains a Smale horseshoe. This implies that such systems possess periodic orbits with arbitrarily high period. The method uses techniques originally due to Melnikov and applies to systems of the form x=fo(X)+εf1(X,t), where fo(X) is Hamiltonian and has a homoclinic orbit. We give an example from structural mechanics: sinusoidally forced vibrations of a buckled beam.

Mechanical models for insect locomotion: dynamics and stability in the horizontal plane I. Theory.

Abstract. We study the dynamics and stability of legged locomotion in the horizontal plane. Motivated by experimental studies of insects, we develop two- and three-degree-of freedom rigid body models with pairs of ‘virtual’ elastic legs in intermittent contact with the ground. We focus on conservative compliant-legged models, but we also consider prescribed forces, prescribed leg displacements, and combined strategies. The resulting mechanical systems exhibit periodic gaits whose stability characteristics are due to intermittent foot contact, and are largely determined by geometrical criteria. Most strikingly, we show that mechanics alone can confer asymptotic stability in heading and body orientation. In a companion paper, we apply our results to rapidly running cockroaches.